Formulas for the skin effect

circuits of the second amplifier stage the condenser C2can be made to introduce a voltage El across thecathode resistor of this stage that exactly balances outthe voltage E2 applied to the grid of the same tubebecause of the presence of voltage across the commonplate impedance Z,.Let Eo=voltage across the common plate imped-

anceE,=voltage introduced across the cathode of

second amplifier stage due to the presenceof Eo

E2= voltage existing on the grid of second ampli-plifier stage due to the presence of Eo

X = 2wrX frequency.Then,

1Ef1=Et0 (1 '''*)E, Eo

C3

C2 frC2R3

E2 = Eo Z-

(2)1+-R1?, R, Rp_1+ + +

R2 RP jcoC,R2

where,Rp = plate resistance of the input amplifier tube

For perfect neutralization, the voltages E, and E2should be equal in magnitude and in phase, which gives

C3 R, / R3

_-=-( 1+-- (3)C2 R2 RpC, R3 / R, \

-Rt1+ ). ~~~~(4)C2 R2 REquations (3) and (4) are independent and both mustbe satisfied. It will be noted that the conditions forbalance are independent of frequency.

This neutralizing circuit reduces power-supply humto the same extent as it does regeneration. This is be-cause such hum is caused by a voltage fed back fromthe power supply to the input stages, just as regenera-tion is caused by a voltage fed back in the same way.The only difference in the two cases is that the voltageacross the output of the power supply arises from dif-ferent causes.

Formulas for the Skin Effect*HAROLD A. WHEELER t, FELLOW, I.R.E.

Summary-At radio frequencies, the penetration of currentsand magnetic fields into the surface of conductors is governed by theskin effect. Many formulas are simplified if expressed in terms of the"depth of penetration," which has merely the dimension of length butinvolves the frequency and the conductivity and permeability of theconductive material. Another useful parameter is the "surface resistiv-ity" determined by the skin effect, which has simply the dimension ofresistance. These parameters are given for representative metals by aconvenient chart covering a wide range offrequency. The 'incremental-inductance rule" is givenfor determining not only the effective resistanceof a circuit but also the added resistance caused by conductors in theneighborhood of the circuit. Simpleformulas are given for the resistanceof wires, transmtssion lines, and coils; for the shielding effect of sheetmetal;for the resistance caused by a plane or cylindrical shield near acoil; andfor the properties of a transformer with a laminated iron core.T HE "skin effect" is the tendency for high-

frequency alternating currents and magneticflux to penetrate into the surface of a conductor

only to a limited depth. The "depth of penetration"is a useful dimention, depending on the frequencyand also on the properties of the conductive material,its conductivity or resistivity and its permeability. Ifthe thickness of a conductor is much greater than thedepth of penetration, its behavior toward high-fre-quency alternating currents becomes a surface phe-nomenon rather than a volume phenomenon. Its

*Decimal classification R144XR282.1. Original manuscript re-ceived by the Institute, May 13, 1942. Presented, Rochester FallMeeting, November 10, 1941.

t Hazeltine Service Corporation, Little Neck, N. Y.

"surface resistivity" is the resistance of a conductingsurface of equal length and width, and has simply thedimension of resistance. In the case of a straight wire,the width is the circumference of the wire.

Maxwell' discovered that the voltage required toforce a varying current through a wire increases morethan could be explained by inductive reactance. Heexplained this as caused by a departure from uniformcurrent density. This discovery was followed up byHeaviside, Rayleigh, and Kelvin. It came to be calledthe "skin effect," because the current is concentratedin the outer surface of the conductor. The ratio ofhigh-frequency resistance to direct-current resistancefor a straight wire was computed in terms of Besselfunctions and was reduced to tables.2-7

I J. C. Maxwell, "Electricity and Magnetism," on page 385.1873/1937, vol. 2, section 690, p. 322.

2 Lord Rayleigh, Phil. Mag., vol. 21, p. 381; 1886.3 C. P. Steinmetz, 'Transient Electric Phenomena and Oscil-

lations," pp. 361-393, 1909/1920.4 S. G. Starling, "Electricity and Magnetism," 1912/1914, pp.

364-369.' E. B. Rosa and F. W. Grover, "Formulas and tables for the

calculation of mutual and self-inductance (revised)," Bureau ofStandards, S-169, pp. 172-182, 1916.

6 "Radio Instruments and Measurements," Bureau of Standards,C-74, pp. 299-311, 1918/1924.

7 J. H. Morecroft, "Principles of Radio Communication," 1921,pp. 114-136.

Proceedings of the I.R.E.412 Sebtember, 1942

Wheeler: Formulas for the Skin Effect

Steinmetz defined the "depth of penetration" with-out restriction as to the shape of the conductor. Heapplied this concept to laminated iron cores, as wellas to conductors Unfortunately, he gave two definitionswhich differ slightly, one for iron cores and another forconductors. The latter definition has been generallyadopted, as in the Steinmetz tables on page 385.More recent writers have reduced the treatment

of the skin effect to simple terms and have general-ized its application.8-'6 Schelkunoff and Stratton havegiven the most comprehensive treatment of the sub-ject, including the depth of penetration in all kinds ofproblems involving conductors. They have introducedthe concept of surface impedance, from which thesurface resistivity is a by-product.

In spite of this active history of the skin effect, thereis still a need for a simple and direct summary whichwill facilitate its appreciation and its application to sim-ple problems. That is the purpose of this presentation.

Following Harnwell and Stratton, the mks ra-tionalized system of units is employed for all relations,except where inches are specified. The properties ofmaterials are taken for rocm temperature (20 degreescentigrade or 293 degrees absolute). The following listgives the principal symbols used herein.

d =depth of penetration (meters)Ri=surface resistivity (ohms)-= conductivity (mhos per meter)p= 1/o-= resistivity (ohm-meters),=permeability (henrys per meter)yo = 47r 10-7 = permeability of spacef=frequency (cycles per second)co = 27rf= radian frequency (radians per second)e=2.72 =base of logarithms

exp x= ex = exponential functionZ=depth from the surface into the conductive

medium (meters)w = width (meters)I= length (meters)

8 E. J. Sterba and C. B. Feldman, "Transmission lines for short-wave radio systems," PROC. I.R.E., vol. 20, PP. 1163-1202; July,1932; Bell Sys. Tech. Jour., vol. 11, pp. 411-450; July, 1932. (Con-venient formulas.)

I S. A. Schelkunoff, "The electromagnetic theory of coaxialtransmission lines and cylindrical shields," Bell Sys. Tech. Jour.,vol. 8, pp. 532-579; October, 1934. (The most complete theoreticaltreatment.)

10 S. A. Schelkunoff, "Coaxial communication transmissionlines," Elec. Eng., vol. 53, pp. 1592-1593; December, 1934. (A briefdescription of the physical behavior.)

"1 E. I. Green, F. A. Leibe, and H. E. Curtis, "The proportioningof shielding circuits for minimum high-frequency attenuation,"Bell Sys. Tech. Jour., vol. 15, pp. 248-283; April, 1936.

12 August Hund, "Phenomena in High-Frequency Systems,"1936, pp. 333-338.13 S. A. Schelkunoff, "The impedance concept and its application

to problems of reflection, refraction, shielding and power absorp-tion," Bell Sys. Tech. Jour., vol. 17, pp. 17-48; January, 1938.

14 G. P. Harnwell, "Principles of Electricity and Electromag-netism," pp. 313-317, 1938.

5 W. R. Smythe, "Static and Dynamic Electricity," 1939, pp.388-417.16 J. A. Stratton, "Electromagnetic Theory," pp. 273-278, 500-

511, and 520-554, 1941. (mks units.)

a= thickness or radius (meters)b =distance, length or width (meters)c = distance (meters)r=radius (meters)A= area (square meters)I= current (amperes)i = current density at a depth z (amperes per

square meter)io=current density at the surface (z=0)H= magnetic field intensity at a depth z (amperes

per meter)H0= magnetic field intensity at the surface (z =0)E =electromotive force (volts)P = power (watts)P1 = power dissipation per unit area (watts per

square meter)Z=R+jX=impedance (ohms)X = reactance (ohms)R = resistance (ohms)G = conductance (mhos)L =inductance (henries)Lo= inductance in space outside of conductive

mediumm =number of laminationsn=number of turnsr = ratio of resistivityx=ratio of radii

Q=ratio of reactance to resistanceFig. 1 is a chart'7 giving the surface resistivity R,

and the depth of penetration d for various metals, overa wide range of frequency f. The depth is plotted inparts of an inch, since this aids in practical applicationand introduces no confusion with the mks electricalunits. Each sloping line represents one metal, depend-ing on its resistivity p or conductivity o- and its per-meability ,u at room temperature (20 degrees centi-grade or 293 degrees absolute). The heavy lines arefor copper, which is the logical standard of comparison.Additional lines can be drawn to meet special re-quirements, shifting them from the copper line inaccordance with the properties of the metal.

Fig. 2 shows a slab of conductive material to be usedin describing the skin effect. The current I is concentrated in the upper surface. From Harnwell, the al-ternating-current density i in the surface of aconductor decreases with depth z according to theformula

=exp-z jIOt00

/wto= exp -(1 +j)zV 2

z z= exp--exp - jd d (1)

17 This chart has been reprinted in the report of the RochesterFallTMeeting in Electronics, December, 1941. More recently,a similar chart has appeared in the following reference, togetherwith other valuable formulas and curves: J. R. Whinnery, "Skineffect formulas," Electronics, vol. 15, pp. 44-48; February, 1942,

413

10o-I

_

il

,I

I1111111If

II

II

1 rlln

Iinc

h

(M)copper

(2)aluminum

(3)br

oss

(4)monga

nin

f-fr

eque

ncy

(cycles)

(5)s

oft

iron

forsm

all

(6)tronsformer

iron

,magnetic

(7) p

ermalloy

78in

tens

ity

Fig.

1-Su

rfac

eResistivityandDe

pth

ofPenetration.

011 coE 0 -._ 4._ 0 U. oo cs 0 va 4-fL.. If

Imm)

_ 0-C,

-2

c C 0 a I- 4S.-

ml)

0c,Q._

-

44X 0

-6C

-4

0.-0 II

A'a*

c

C% ;t


This decay of current density is shown by the shadedarea plotted on the side of the slab.The depth of penetration is defined by the last

formula, as the depth at which the current density (ormagnetic flux) is attenuated by 1 napier (in the ratio1/e= 1/2.72, or -8.7 decibels). At the same depth, itsphase lags by 1 radian, so d is 1/27r wavelength or 1radian length in terms of the wave propagation inthe conductor.The depth of penetration, by this definition, is

2r 1d = X-

Cwy -\rfma

E piZ = = (1 +j)

I wd

= (1 + j)-1 1=\/_Pw w

ohms. (5)

Its real and imaginary components are the resistance

in spLice,u0meters. (2) (sqrface)--

It is noted that the v/2 factor arises when the /Vj isresolved into its real and imaginary components in theexponent in (1).The total current is the integral of the current den-

sity in the conductive medium. This integral from thesurface into the medium is a decaying spiral in thecomplex plane, which rapidly approaches its limit ifthe thickness is much greater than the depth of pene-tration. The total current is therefore given by theintegral for infinite depth, over the width w:

fxI=W i.dzGo

=iowJ exp- (l+j)-dzo ~~~d

iowd1 + j amperes. (3)

The voltage E on the surface along the length ofthe conductor is obtained from the current density and

1~~~~~~~Fig. 2-The skin effect on the surface of a conductor.

the volume resistivity.E = iolp volts. (4)

If this voltage were to be measured, the return circuitwould have to be adjacent to the surface so as not toinclude any of the magnetic flux in the near-by space.The "internal impedance" or "surface impedance"

is computed from this voltage E and the current I.

in conductoru,pe anda

(a) (b)

I I__._I __---- _.._. _. _L(center of symnietrical conductor, or opposite surface

of shieldiing partition.)Fig. 3-The internal impedance of a conductor, in terms of dis-

tributed circuit parameters (a) and equivalent lumped parame-ters (b).

and the internal reactance, which are equal.Z = R + jX,

IR= X=- 7rf/p

wohms. (6)

The surface resistivity R1, given in the chart, isdefined as the resistance of a surface of equal lengthand width.

R1= =;f=d /Tp

IR= X = R

wohms. (7)

For example, R1 is the resistance of the unit squaresurface in Fig. 2.18The internal inductance is the part of the total in-

ductance which is caused by the magnetic flux in theconductive medium. It is computed from the internalreactance.

X lI d\L =-=

-IA--co w\2/ henries. (8)

This is the inductance of a layer of the conductivematerial having a thickness of d/2, one half the depthof penetration. This merely means that the meandepth of the current is one-half the thickness of theconducting layer.

Fig. 3 illustrates the concept of internal impedancein terms of electric circuit elements. In the diagram(a), the inductance Lo is that caused by the magneticflux in the space adjacent to the conductor. Each part

18 Schelkunoff (footnote 9, p. 550) calls R1 the "intrinsic re-sistance" of the material.

1942 415

Proceedings of the I.R.E.

of the current meets additional inductance in propor-tion to its depth from the surface of the conductor.This inductance is AL per element of depth. The con-ductance of the material is AG for the same elementof depth. The conductive slab behaves as a trans-mission line with paths of shunt conductance in layersparallel to the surface, and series inductance betweenlayers. This hypothetical line presents the internalimpedance Z in series with the external inductanceLo. If the thickness of the conductor is much greaterthan the depth of penetration, the impedance is un-affected by conditions at the far end of the line, orbeyond the other surface of the conductor.The internal impedance of the hypothetical trans-

mission line in Fig. 3(a) is computed from its dis-tributed inductance and conductance, by circuittheory. Since the magnetic flux path has an arealAz and a length w

,ulA\zAL = henries. (9)

w

Since the current path has an area wAz and a length 1,uwAz wAz

AG = = mhos. (10)I p1

The impedance of a long line with these properties is..

Z= NjAL/AG=-jp ohms. (11)w

This is an independent complete derivation of (5),without recourse to electromagnetic-wave equations.The components of internal impedance are shown in

Fig. 3(b) as R and L. The resistance R is that of a layerwhose thickness is equal to the depth of penetration d.The internal inductance is that of a layer whose thick-ness is d/2, one half the depth of penetration.Some inductance formulas carry the assumption

that the current travels in a thin sheet on the surfaceof the conductor, as if the resistivity were zero. Suchassumptions are usual for transmission lines, waveguides, cavity resonators, and piston attenuators. Suchformulas can be corrected for the depth of penetrationby assuming that the current sheet is at a depth d/2from the surface. This is the same as assuming that thesurface of the conductor recedes by the amount

d,. (12)

2 yoThe second factor has an effect only if the conductivematerial has a permeability ,t differing from that ofspace pto. The same correction is applicable to shieldingpartitions, regarding their effect on the inductance ofnear-by circuits.There is sometimes a question which surface of a

conductor will carry the current. The rule is, that thecurrent follows the path of least impedance. Since the

impedance is mainly inductive reactance, in the com-mon cases, the current tends to follow the path of leastinductance. In a ring, for example, the current densityis greater on the inner surface. In a coaxial line, thecurrent flows one way on the outer surface of the innerconductor and returns on the inner surface of the outerconductor.

In determining whether the thickness is muchgreater than the depth of penetration, the effectivethickness corresponds to the length of the hypotheticalline in Fig. 3(a). In a symmetrical conductor withpenetration from both sides, as in a strip or a wire, theeffective thickness is the depth to the center of the con-ductor. In a shielding partition with penetration intothe surface on one side and with open space on theother side, the effective thickness is the actual thick-ness. If the effective thickness exceeds twice the depthof penetration, the accuracy of the above impedanceformulas is sufficient for most purposes, within twoper cent for a plane surface.The shielding effect of a conductive partition de-

pends not only on the material and thickness of thepartition, but also on its location. For example, twolayers of metal have more shielding effect if they areseparated by a layer of free space than if they are closetogether. If a shielding partition carries current on onesurface (z = 0) and is exposed to free space at the othersurface (z =a) the current density has a definite ratiobetween one surface and the the other. For the thick-ness a, much greater than the depth of penetration, asin Fig. 2, this ratio is

ta a-= 2 exp --jo d

a= 0.69 -

da

= 6- 8.7d

(a >> d)

napiers

decibels. (13)

The factor 2 is caused by reflection at the far surface.The space on either side of a shield usually adds to theattenuation indicated by this formula.The shielding ability of a given metal at a given

frequency is best expressed as the attenuation for aconvenient unit of thickness, disregarding the reflectionfactor. The unit of thickness may be 1 millimeter(10-3 meter) or 1 mil (2.54- 10-5 meter). In copper at1 megacycle, for example, it is 132 decibels per milli-meter or 3.3 decibels per mil. In iron, it is much greaterand depends also on the magnetic flux density, sincethat affects the permeability.The power dissipation in the surface of a shield is de-

termined by the magnetic field intensity at its surface.The same is true of current conductors or iron coresbut in those cases there are more direct methods ofcomputation in terms of current and effective resist-ance. Since the magnetic flux path has a length equal to

416 Sepbtember


the width w of the conductor, and since the magneto-motive force is equal to the current I, the magneticintensity at the surface is

I

w

(c) Note that the increment of inductance causedby penetration into each surface is

, d aLoL

yo 2 (9amperes per meter. (14)

The power dissipation isI

P = 12R = (wHo)2 R,w

= IwHoI2R = lwP,

henries. (17)

(d) Compute the effective resistance contributed byeach surface,

watts (15)in which the power dissipation per unit area of surfaceis

Pi = Ho2RI watts per square meter. (16)For most purposes, the power dissipation is morereadily computed by the following method, in terms ofeffective resistance in a circuit.The "incremental-inductance rule" is a formula

which gives the effective resistance caused by the skineffect, but is based entirely on inductance computa-tions. Its great value lies in its general validity for allmetal objects in which the current and magnetic in-tensity are governed by the skin effect. In other words,the thickness and the radius of curvature of exposedmetal surfaces must be much greater than the depthof penetration, say at least twice as great. It is equallyapplicable to current conductors, shields, and ironcores.

This rule is a generalization of (7) which states thatthe surface resistance R is equal to the internal re-actance X as governed by the skin effect. The internalreactance is the reactance of the internal inductance Lin (8). This inductance is the increment of the totalinductance which is caused by the penetration of mag-netic flux under the conductive surface. This change ofinductance is the same as would be caused by the sur-face receding to the depth given in (12). Starting witha knowledge of this depth, the reverse process of com-putation gives the increment of inductance caused bythe penetration, and from that the effective resistanceas governed by the skin effect.The incremental-inductance rule is stated, that the

effective resistance in a circuit is equal to the changeof reactance caused by the penetration of magneticflux into metal objects. It is valid for all exposed metalsurfaces which have thickness and radius of curvaturemuch greater than the depth of penetration, say atleast twice as great.The application of the incremental-inductance rule

involves the following steps:(a) Select the circuit in which the effective resistance

is to be evaluated, and identify the exposed metalsurfaces in which the skin effect is prevalent.

(b) Compute the rate of change of inductance ofthis circuit with recession of each of the metal surfaces,o-Lo/az, assuming zero depth of penetration.'9

1Y A second-order approximation is secured if bLo/8z is computed

1 dLoR-c=L = R1

Ho oazohms. (18)

For a surface carrying the current of the circuit, thisis identical with (7). For the effect of near-by metalobjects, such as shields, this formula is easily appliedin many practical cases. It is most useful in cases ofnonuniform current distribution, which otherwisewould require special integrations.A straight wire has its current concentrated in a

tubular surface layer as shown in Fig. 4(a). The depth

(a) r

(a) High-frequency current tube.

(b) 2r- d,

(b) High-frequency mean diameter.

(c) r exp-.u

(c) Low-frequency mean diameter.Fig. 4-The current distribution in a straight wire.

of this layer is d. The radius of the wire is r but themean radius of the current tube is r-d/2. The resist-ance ratio of the wire is the ratio of the alternating-current resistance R of the direct-current resistanceRo. It is the inverse ratio of the effective cross-sectionalareas,

R 7rr2Ro r(2r - d)d

r 12d 4 (r > 2d). (19)

assuming that the surface is below the actual surface by the amountgiven in (12).

1942 417


Since the assumptions are an approximation at best,only the first two terms of this series deserve attention.They give a close approximation if the radius exceedstwice the depth of penetration, or if the resistance ratioexceeds 5/4.20-26The inductance of a straight wire is determined by

the mean diameter of the current path. Fig. 4(b) showsthe equivalent current sheet for the case in which theradius is very much greater than the depth of penetra-tion. A perfect conductor, to have the same inductancewith zero depth of penetration, has a radius which isless by the amount given in (12). This rule is reliableonly if the equivalent radius is greater than 7/8 theactual radius.

Eule~~~~rFig. 5-Straight wire.

The low-frequency inductance of a straight wire,with uniform current distribution, is its maximum in-ductance. As shown in Fig. 4(b), the equivalent cur-rent sheet has the radius

r exp-- (20)4go

in which the factor exp-1/4 is the "geometric-meandistance" of a circular area."The straight wire of Fig. 5, assuming a depth of pene-

tration very much less than the radius, has its resist-ance expressed by the simple formula

IR=

-RI ohms. (21)27rrThis neglects the second term in the series of (19). Itis on this simple basis that the following cases aredescribed.28'8,1'The coaxial line of Fig. 6 has its current flowing one

way on the lesser radius ri and returning on the greaterradius r2. The total resistance is

/1 1\lR = (-+-)-R , ohms. (22)

ri r2 2ir20 Morecroft, (footnote 7, p. 116), curves of resistance ratio.21 E. Jahnke and F. Emde, "Tables of Functions," B. G.

Teubner, Berlin, Germany, 1933, chapter 18, p. 314, Fig. 165, curverb0/2b,.

12 August Hund, "High-Frequency Measurements," 1933. pp.263-266. Series expansions.

23 Schelkunoff, footnote 9, pp. 551-553, formulas and curvesfor resistance and reactance ratio.

24 August Hund, "Phenomena in High Frequency Systems,"1936. p. 338. Series expansions.

25 J. H. Miller, "R -F resistance of copper wire," Electronics,vol. 9, no. 2, p. 338; February, 1936. Curves and formula.26 Stratton, footnote 16, p. 537, series expansions.

27 Rosa and Grover, footnote 5, p. 167.28 Alexander Russell, "The effective resistance and inductanceof a concentric main," Phil. Mag., sixth series, vol. 17, pp. 524-552;April, 1909.

The inductance in the space between the conductorsis29

Lo = log-27r ri

henries. (23)

For a given value of the greater radius r2, minimum at-tenuation in this line requires minimum R/Lo, and thisis obtained with r2/rl=3.59, approximately.30"3, With

Fig. 6-Coaxial conductors.

this shape, the resistance of inner and outer conductorsis divided as 78 per cent and 22 per cent of the total.Since the optimum ratio satisfies the equation

log-r2 r1lg= I + )ri r2

(24)the ratio of reactance to resistance for this shape is, fora nonmagnetic conductor,32

2r, r2Qd= 1.=d 1.8 d (A = go). (25)

This is the ratio of the diameter of the inner conductorto the depth of penetration. In general,

Lor22r, .5rd ri1 +-

r2

(26)

This value is reduced slightly by end effects.If a coaxial line is used as the inductance of a reso-

nant circuit, maximum impedance at parallel resonancemay be desired. This is obtained with maximum Lo2/R,which determines the condition

- log - = 1 + -.2 rj r2 (27)

The required shape is r2/rl = 9.2, approximately.33 If thelength of the line is much less than one-quarter wave-length, so its shunt capacitance is negligible, this opti-mum shape has the following resistance at parallelresonance: (, =yo).

R' = Q2R = 0.307 dr RI ohms. (28)For given frequency and material, this resistance isproportional to the area of the conducting surfaces.

29 Harnwell, footnote 14, p. 304.30 Sterba and Feldman, footnote 8, p. 419.11 Green, Leibe, and Curtis, footnote 11, p. 253.32 In all cases, Q is expressed on the assumption of a nonmag-

netic conductor.33 F. E. Terman, "Resonant lines in radio circuits," Elec. Eng.,

vol. 53, pp. 1046-1053; July, 1934.

418 September


A pair of straight parallel wires is shown in crosssection in Fig. 7. The same current flows in oppositedirections in the two wires, and is concentrated on thesurface. If the wire diameter 2a is much less than thecenter-to-center separation 2b, the resistance of eachwire is given by (21) for Fig. 5. As the diameterin Fig. 7 approaches equality with the separation, theproximity of wires causes greater current density onthe inner sides.34 35 This effect is easily evaluated bythe incremental-inductance rule. The approximate and

ab

Fig. 7-Parallel wires.exact formulas for the external inductance of this pairof wires, of length 1, are

IAl 2bLo=- log - (a


From this formula and (18), the added resistance is3irr24

R'=- R1' ohms1 6c4 (38)

in which R1' is the surface resistivity of the shield. Thisis equal to the change of reactance which would becaused if the shield were moved further back by thedisplacement (d/2) (/u/uo) as shown in Fig. 9. Compar-ing the added resistance with the resistance R of thering alone, formula (32), the relative change of resist-ance is

R' 37rrlr23Rl'R 16c4RI

This ratio is independent of the frequency, so long asthe depth of penetration is the controlling factor. Asan example, a copper ring with the optimum shape

Fig. 9-A ring near a shielding wall.

(r2 = 2.5ri) at a distance of 1 diameter from a soft-ironshield (R1'=40R1) would suffer about 59 per cent in-crease of resistance caused by the shield. In this loca-tion, a slightly smaller wire diameter would be opti-mum, because the inductance of the ring would increasein a greater ratio than the total resistance. The reduc-tion of inductance (36) by the shield varies with theinverse cube of the distance, whereas the added re-sistance (38) varies with the inverse fourth power.A ring perpendicular to the shield, instead of parallel

as in Fig. 9, and with its center at the same distance,would suffer only one half as much change of induct-ance and resistance. This follows from the fact thatthe mutual inductance with its image would be one halfas great. This is a striking example of the utility of theincremental-inductance rule, since the departure fromaxial symmetry would make this problem very difficultof solution by field-integration methods.A coil of n turns near a shield has its inductance and

resistance changed by n2 times as much as the ring,that is, by n2Lo' and n2R', formulas (36) and (38).The air-core toroidal coil of Fig. 10 has n turns on a

coil radius of ri and a ring radius of r2. The followingsimple formulas are based on the asumptions that thecoil radius is much less than the ring radius (rl


There is another optimum design for a given outsidediameter, 2 (r,+ r2):

riri- r2= 0.41;r1 + r2

r2_= 0.59

r1 + r2

inner and outer surfaces of the coil. The theoretical re-lations are based on the ideal long coil and shield,closely wound of rectangular wire, but the conclusionsare approximately correct for practical coils. The mag-netic intensity H1 inside the coil and H2 between coil

- = 0.70 (48)

ri+ r2 rQ = 0.343 = 0.83d d

The solenoidal coil of Fig. 11 has n turns wound on aradius a in an axial length b. If such a coil has a lengthmuch greater than its radius and is wound closely withrectangular wire of thickness much greater than the

n turns____b *

Fig. 11-Solenoidal coil.

depth of penetration, the current flows in a sheet onthe inner surface of the wire, and the resistance is

2iraR =

-n2R,b ohms. (49)

In a practical coil of many turns of round wire, thereis an optimum diameter of wire slightly less than thepitch of winding. This formula is a rough approxima-tion for practical coils with optimum wire diameter.It corresponds to a coil resistance slightly less than -rtimes as great as the resistance of a straight wire of thesame length and diameter. (The effect of distributedcapacitance and dielectric resistance is omitted.) Theinductance is approximately,39 for b > 0.8a

Fig. 12-Solenoidal coil in a coaxial tubular shield.

and shield are in the inverse ratio of the cross-sectionalareas because all the flux inside the coil has to returnin the space between coil and shield.

H2 a,2 1H1 a22

-a,2 a22/a12 - 1 (52)

The power dissipation is divided among the inner andouter surfaces of the coil and the inner surface of theshield. By (15), the total isP = 27ra,bH12Rl + 27ra,bH22R, + 27ra2bH22R2

2 ( 22 a22H22R2)= 2,7raibHl R,l 1 + -+1

\ H2 a2H 2Rwatts (53)

in which R1 and R2 are the respective values of surfaceresistivity for the metals of coil and shield. By (14),the total current on both surfaces of the coil is

I = (H1 + H2)b/n amperes. (54)

= oara2n2Lo =

b + 0.9 ahenries. (50)

The corresponding ratio of reactance to resistance isapproximately

a IQ= -d 1 + 0.9 a/b (51)

These simple formulas are applicable to coils in whichthe length is greater than the radius, the optimum wirediameter exceeds 4d, and the number of turns exceedsabout 4. In comparison with some recent measurements,these formulas check fairly well the component ofresistance caused by the skin effect as distinguishedfrom capacitance effects.40A solenoidal coil in a coaxial tubular shield is shown

in Fig. 12. The radius of coil and shield a, and a2 de-termines the relative distribution of current on the

39H. A. Wheeler, "Simple inductance formulas for radio coils,"PROC. I.R.E., vol. 16, pp. 1398-1400; October, 1928.

40 F. E. Terman, "Radio Engineering," 1932/1937, pp. 37-42.

Therefore, the effective resistance of the coil isp

R = 122aR(+H22 a22H22R2)

2 7raln'Ri H12 a12H12R,+ \

\bH

b b

(55)

The last factor gives the effect of the shield. It mayactually reduce the resistance, by redistribution ofsurface currents, but not as much as it reduces theinductance. The effective inductance of the coil in theshield is

+

ira12 r(a22-a12)

=Iral xI(a2-7ral2Po a,2\

9-henries (56)

4211942


in which the last factor gives the reduction of induct-ance by the shield. With these substitutions,

a1 Rx= , r=- (57)

a2 R,and the ratio of reactance to resistance is

a2 x(j -x2)Q = _. . (58)d 1-(2--r)x2+2x4 (

This is expressed in terms of the shield radius a2 sincethat determines the space in which the coil is located.The maximum Q is obtained if x satisfies the equation

0 = 1-(1 + r)x2-(4 + r)x4 +x6. (59)This is most easily solved by trial. The solutions for theoptimum design in several cases are as follows:

R2 a,r=- X2 X=-

R, a2

0 0.41 0.64

1 0.30 0.55

2 0.23 0.48

1 1r \/r

Q (60)

is not accompanied by a proportionate reduction of Q.Therefore, the shield reduces the effective resistance inthese cases. In practice, the shield always decreases theQ.The transformer of Fig. 13 has a laminated iron core

of cross-sectional area A. The flux path in the iron hasa length 1i while that in the air gap has a length 1,. Forsimplicity of analysis, the two coils have the same num-ber of turns n. If the actual number of turns is ni andn2, the respective self-impedances and mutual imped-ance are obtained by letting

n2 = n12, n22, nln2. (62)Fig. 14 shows the impedance network which is the

equivalent of this transformer. The upper part repre-sents the coil resistance and the part of the inductancecaused by magnetic flux in the space outside the core,as if the core space had zero permeability. The lower

r__ _ _ _ _ _ _ __inair,outside of core Ia2 a,0.72 - = 1.14-d da2 a,0.44 - = 0.80d da2 a10.33 - = 0.70-d d

1 a2 a,_- = 0.502\/r d d

An approximate formula for the optimum ratio of radiiis given by the relation,

1X2 =

2.3 + ra, 1a2 V2.3 + R2/R1

(61)

This formula is exact for r= 1, 2, oo. In the first tworows of the table (60), the coefficient in the last columnindicates that the reduction of inductance by the shield

)07n turnln turne

TA rn~a

bFig. 13-Transformer with laminated iron core.

Fig. 14-The distributed-impedance network equivalentto the iron-core transformer.

part represents the impedance caused by the core,including the air gap. The inductance which wouldbe caused by the flux in the iron core of permeability u,with no air gap, is

1un2ALi = An2t

lihenries. (63)

The inductance which would be caused by the flux inthe air gap, if the iron core had infinite permeability, is

gOn2A henries. (64)

The inductance effective at low frequencies is that ofLi and L. in parallel,

LILt,, yo2ALo= henries. (65)

Li + Lg, ig + li-to/,4The eddy currents and skin effect depend on the di-vision of the core area into laminations,

A = mab square meters (66)in which m is the number of laminations of thickness aand width b. The current paths in the laminations

422 September


cause an apparent distributed conductance, associatedwith the iron inductance Li, which has the value

Gi = mhos (67)4n2mb

in which o- is the conductivity of the iron. The effect ofthis distributed conductance is least at low frequenciesand merges into the skin effect at high frequencies.

Fig. 15 shows a simplified equivalent network inwhichthe shunt resistance R and inductance L have valuesdepending on the frequency. These parallel componentsare used rather than series components, because theeffective shunt resistance varies less with frequencythan the effective series resistance.At low frequencies, the apparent shunt conductance

approaches the constant valueG = iGi mhos. (68)

The corresponding value of shunt resistance is12n2mbd

R= Rali

ohms. (69)

in which R, is the surface resistivity of the iron. Theinductance L has its low-frequency value Li. This isbased on the assumption that the alternating fluxwithin the lamination suffers only a small phase lagand no appreciable reduction in magnitude, which istrue if the depth of penetration is greater than thethickness of laminations.4' The corresponding ratio ofshunt susceptance to conductance, is

R d2Q = =6-wL a2 (d >> a). (70)

At frequencies so high that the depth of penetrationis less than 1/4 the thickness of laminations, the skineffect governs the impedance caused by the iron core.The effective impedance of R and L in parallel is theimpedance of the line with distributed series Li andshunt Gi:

z 1 L1/R + 1/jwL G4n2mb

=- R ohms. (71)(1 - j)liThe shunt components of this impedance have thevalue

4n2mbR = coL - R ohms. (72)

IiThe apparent shunt inductance is

2dL = -L

ahenries. (73)

This is the inductance based on twice the depth ofpenetration as the effective thickness of each lamina-tion.The air gap sometimes increases the ratio of react-41 V. E. Legg, "Survey of magnetic materials and applications in

the telephone system," Bell. Sys. Tech. Jour., vol. 18, pp. 438-464,July 1939. (In Fig. 7, 0 is the ratio of thickness to depth of pene-tration.)

ance to resistance in the impedance of an iron-coreinductor. This question involves the series resistance Rcof each coil, while the inductance in the space outsidethe coil is usually negligible. Increasing the air gap

Fig. 15-The lumped-impedance network equivalentto the iron-core transformer.

decreases Lq,, thereby causing more dissipation in thecoils (R,) and less in the core (R). The optimum lengthof air gap is approximately that which divides thedissipation equally between coil and core. The opti-mum condition is

1 1 /1 + R,/R =,'/

WL wL, RR,For this condition, the maximum ratio is42

1/ R/RCQ2= V2 V1 + R,I/R

(74)

(75)This is nearly independent of the number of turns. Itsvalue is expressed in terms of three properties of thecoil; Pc is the resistivity of the copper wire, l4 is theaverage length of wire per turn, and A, is the totalcross-sectional area of the turns of wire on the windingin question. In a self-inductor of one coil, A, is some-what less than the area of each window. The followingformulas are simplified on the assumption that R>>R,so the optimum air gap gives Q>> 1. At the higher fre-quencies, where the skin effect predominates, theoptimum air gap gives

AAcpadlil,p, (a > 4d). (76)

At the lower frequencies, where eddy currents areinduced by nearly uniform flux in the laminations, theoptimum air gap gives_3

V 3AA pa2liJcpc (a < d). (77)

These maximum values cannot be realized if the opti-mum length of air gap is negative. This is true at verylow frequencies where the eddy currents are negligible,in which case the air gap is reduced to zero, giving

wL~Q = toLi (wLi


and copper, the materials of the core and the coil. Inthese formulas, the depth of penetration includes thefrequency dimension, enabling the expression entirelyin terms of ratios. The value of (78) is actually inde-pendent of the p of the iron and the iuo of the copper,those being involved also in the depth of penetration.

Since copper is the usual material for conductors,it is useful to remember the depth of penetration incopper at a certain frequency and room temperature(20 degrees centigrade):At f 106 cycles=1 Mc,

= 66 10-6 meter = 0.066 mm = 66 microns= 2.6 10-3 inch = 2.6 mils. (79)

The values for copper and other materials (at a tem-perature of 0 degrees centigrade) are found in the Stein-metz3 table, p. 385. The essential properties of copperare (at 20 degrees centigrade):

Pe = Io = 47r 10-7 henry per meter= 1.257 microhenrys per meter

c = 5.80 107 mhos per meter= 58 megamhos per meter

Pc = 1.724. 10-8 ohm-meter= 1/58 microhm-meter

Aooc = 72.8 seconds per square meter

(80)

d and the surface resistivity R1 plotted against fre-quency. Each pair of crossed lines is for one material.Some of the materials shown are chosen for their ex-treme properties (at least, among the common ma-terials). Copper has the least resistivity. The permalloyshown (78 per cent nickel) is used for loading subma-rine telegraph cables and for shielding against alter-nating magnetic fields; it has the least depth of pene-tration, by virtue of its high permeability and smallresistivity:

Pt = 9000 ,0 (at small flux density)p = 9.3 pc = 0.16 microhm-meterAt 1 Mc (83)d' = 2.1 10-6meter = 0.084 mil

Ri' = 75 milohmsManganin is the material usually used in resistancestandards; it has about the highest resistivity com-patible with minimum permeability, and therefore thegreatest depth of penetration:

A = /0op = 25.5 Pc = 0.44 microhm-meterAt 1 Mcd' = 0.33 mm = 13 milsR = 1.3 milohms.

(84)

i-oPc = 2.17 10-14 ohm2-secondin which uo is the permeability of space. The otherimportant value for copper is the surface resistivity,still at 1 megacycle:

R1,1 = 2.60 10-4 ohm= 0.260 milohm. (81)

In order to convert d, and Ri, for other materials, it isnecessary to know only their permeability and resistiv-ity relative to copper:

/ 1 Mc IA0 pd = dc,4f u Pc

R, =RIcA/ --h . (82)V 1 Mc Ao Pc

The chart of Fig. 1 gives the depth of penetration

Most of the ordinary materials fall within the limitsof these three cases.On the chart, the intersection of each pair of lines

moves upward with increasing resistivity and towardthe left with increasing permeability. (It is purelycoincidental that the intersection is at 1 megacyclefor nonmagnetic materials.)

In this collection of formulas, the properties of theconductive materials are usually expressed in terms ofdepth of penetration d and surface resistivity R1, bothof which involve also the frequency. The former ap-pears in ratios with other "length" dimensions. Thelatter appears in impedance formulas, where it bringsin the "resistance" dimension. Other quantities usuallyappear in ratios so they do not complicate the dimen-sions. The two parameters d and R1 are most usefulbecause they have not only dimensional simplicitybut also obvious physical significance.

424 September

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Formulas for the skin effect